Measure Graphs

A bipartite graph with vertex sets X and Y is a triple {X, Y; E), where £ is a subset of X x Y, and a graph with vertex set X is a pair (X; £), where £ is a symmetric subset of X x X minus the diagonal. Nash-Williams [15] proposed the following generalization of these concepts. Let {X,$4,\x) and (Y, 08, v) be measure spaces. A bipartite measure graph with vertex sets X and Y is a triple (X, Y; M), where M is a measurable subset of X x Y with the measure /* ® v. A measure graph with vertex set X is a pair (X\ M), where M is a symmetric measurable subset of X xX minus the diagonal with the measure n ® \i. Nash-Williams [15] (see also [1; p. 101]) asked for measure graph versions of various standard theorems in graph theory, including Turan's theorem [18] on complete graphs, the theorem of Erdos 'and Gallai [6] on degree sequences and Hall's theorem [10] on transversals. The main aim of this note is to investigate the possibility of such extensions. Throughout the note (X, stf, /i) and (Y, (%, v) are atomless complete measure spaces of total measure 1. For simplicity we usually write X and Y for these spaces. Furthermore, the product X x Y is considered to be endowed with the product measure n ® v and X x X with the measure ^ ® fi. If there is no danger of* confusion, we write \U\ for the measure of a set U. The same notation is used for the number of elements of a finite set. Given a set S <= X x Yand x0 e X, y0 e Y, we put